1. Logarithmic and Exponential Functions
Chapter 12
Logarithmic and Exponential Functions

2. The Logarithmic Function
§12.2
The Logarithmic Function

3. Logarithms
The logarithm, base b, of a positive number x is the power (exponent) to which the base b must be raised to produce x. That is, y = logbx is the same as x = by, where b > 0 and b  0.
Exponent = y
y = logbx
x = by
Base = b

4. Exponential Equations in Log Form
Example:
Write the following exponential equations in logarithmic form.
Exponential Form
Logarithmic Form
23 = 8
log28= 3
50 = 1
log51= 0

5. Log Equations in Exponential Form
Example:
Write the following logarithmic equations in exponential form.
Logarithmic Form
Exponential Form
log232= 5
25 = 32
log1010= 1
101 = 10

6. Solving Logarithmic Equations
Example: Solve the following logarithmic equation.
a) logx 81 = 4
b) log10 x = 2
a) logx 81 = 4
x4 = 81
Convert into an equivalent exponential equation.
x4 = 34
Solve the exponential equation.
x = 3
b) log10 x = 2
10– 2 = x
Convert into an equivalent exponential equation.
Simplify.

7. Graphing a Logarithmic Function
Example: Graph y = log2 x.
y = 2x y x 2 –2 2
The exponential form of the function is x = 2y. 2 4 1 1 2 y x
y = log2 x
y = log2 x is the inverse function of y = 2x.

8. Properties of Logarithms
§12.3
Properties of Logarithms

9. The Logarithm of a Product
Property 1: The Logarithm of a Product
For any positive real numbers M and N and any positive base b  1,
logbMN = logbM + logbN.
Example: Write the following logarithm as a sum of logarithms
a) log5(4 · 7) b) log10(100 · 1000)
a) log5(4 · 7)
= log54 + log57
b) log10(100 · 1000)
= log log101000
= = 5

10. The Logarithm of a Product
Example: Write the following logarithms as single logarithms
a) log24 + log2x b) log 50 + log x + log 2
a) log24 + log2x
= log24x
b) log 50 + log x + log 2
= log 50(x)(2)
= log 100x

11. The Logarithm of a Quotient
Property 2: The Logarithm of a Quotient
For any positive real numbers M and N and any positive base b  1,
Example: Write the following as a difference of logarithms.

12. The Logarithm of a Quotient
Example: Write the following as a single logarithm.

13. The Logarithm of a Number Raised to a Power
Property 3: The Logarithm of a Product
For any positive real numbers M, and any real number p, and any positive base b  1,
logbMp = plogbM.
Example: Write the following as a single logarithm.
a) 3 log8 5  log8 z b) 2 log log 10
a) 3 log8 5  log8 z
= log8 53  log8 z
= log8 53  log8 z
b) 2 log log 10
= log log 103

14. Solving Logarithm Equations
The following properties are true for all positive values of b  1, and all positive values of x and y.
Property 4 logb b = 1
Property 5 logb 1 = 0
Property 6 If logb x = logb y, then x = y.
Example: a) Evaluate log9 9.
b) Evaluate log8 1.
c) Find y if log4 y = log4 19.
a) log9 9 = 1
b) log8 1 = 0
c) If log4 y = log4 19, then y = 19.

15. Solving Logarithm Equations
Example: Solve for x.
log5 1 = log5 x – log5 8
log5 1 = log5 x – log5 8
log5 1 + log5 8 = log5 x
Isolate the variable.
0 + log5 8 = log5 x
Property 5
8 = x
Property 6

16. Solving Logarithm Equations
Example: Solve for x.
log3(4x + 6) – log3(x – 1) = 2
log3(4x + 6) – log3 (x – 1) = 2
Property 1
Convert to exponential form.
Simplify.
Solve for x.

17. Exponential and Logarithmic Equations
§12.5
Exponential and Logarithmic Equations

18. Solving Logarithmic Equations
When solving logarithmic equations, we generally try to get all of the logarithms on one side of the equation and the numerical values on the other side. Then the properties of logarithms are used to obtain a single logarithmic expression on one side.
Step 1: If an equation contains some logarithms and some terms without logarithms, try to get one logarithm alone on one side and one numerical value on the other.
Step 2: Convert to an exponential equation using the definition of a logarithm.
Step 3: Solve the equation.

19. Solving Logarithmic Equations
Example:
Solve log24 + log2(x – 1) = 5.
log24 + log2(x – 1) = 5
log2[4(x – 1)] = 5
Property 1
log2(4x – 4) = 5
Simplify.
4x – 4 = 25
Write in exponential form.
4x – 4 = 32
Solve for x.
4x = 36
Check:
x = 9
log24 + log2(9 – 1) = 5
log24 + log28 = 5
2 + 3 = 5 

20. Solving Logarithmic Equations
Example:
Solve log(x + 3) + log x = log 4.
log(x + 3) + log x = log 4
log(x + 3)x = log 4
Property 1
(x + 3)x = 4
Property 6
x2 + 3x = 4
Simplify.
x2 + 3x – 4 = 0
Solve for x.
Stop! It is not possible to take the logarithm of a negative number.
(x + 4)(x – 1) = 0
Check:
x = – 4 or x = 1
log (4 + 3) + log(4) = log 4
log (1 + 3) + log 1 = log 4
log = log 4 

21. Solving Exponential Equations
Property 7
If x and y > 0 and x = y, then logbx = logby, where b > 0 and b  1.
Notice that this is the reverse of Property 6.
Property 7 is also referred to as “taking the logarithm of each side of the equation.”
Example:
Solve 5x = 23.
5x = 23
log 5x = log 23
Property 7
x log 5 = log 23
Property 3
Divide both sides by log 5.

22. Solving Exponential Equations
Example:
Solve 153x – 2 = 230.
153x – 2 = 230
log 153x – 2 = log 230
Property 7
(3x – 2)log 15 = log 230
Property 3
Divide each side by log 15.
3x – 2 =
Simplify.
3x =
Solve for x.
x  1.336

23. Solving Applications Example:
In 2000, a forest had 300 male deer. The growth in the number of deer is estimated by the function g(t) = 300e0.07t where t is the number of years after How many deer will be in the forest in
a) 2010?
b) 2020?
In the year 2000, t = 0.
(Notice that f(0) =300e0.07(0) = 300e0 = 300, the original number of deer.)
In the year 2010, t = 10.
g(10) = 300e0.07(10) = 300e0.7  300( )  604 deer 2010.
In the year 2020, t = 20.
g(20) = 300e0.07(20) = 300e1.4  300(4.0552)  1217 deer 2010.